论文标题
退化多凸能的临界点
Critical points of degenerate polyconvex energies
论文作者
论文摘要
我们研究临界和静止的，即相对于内部和外部变化，$ f（x）= g（\ det（x））$的polycovex函数的关键，对于$ x \ in \ mathbb {r}^{2 \ times times 2} $。特别是，我们表明唇lip（ω，\ mathbb {r}^2）$的关键点$ u \带有$ \ det（du）\ neq 0 $ a.e.除了相对封闭的零尺寸零以外，具有局部恒定的决定因素，并且固定点几乎在任何地方都具有恒定的决定因素。这是从更一般的结果中得出的，这些结果$ u \ in Lip（ω，\ mathbb {r}^n）$，$ω\ subset \ subset \ mathbb {r}^n $ to线性化问题$ curl（βdu）= 0 $。我们还对原始结果进行了一些概括，并假设解决方案$ u $的进一步规律性。最后，我们表明，与上述差异相对于平稳性相关的差分包含是刚性的。
We study critical and stationary, i.e. critical with respect to both inner and outer variations, points of polyconvex functionals of the form $f(X) = g(\det(X))$, for $X \in \mathbb{R}^{2\times 2}$. In particular, we show that critical points $u \in Lip(Ω,\mathbb{R}^2)$ with $\det(Du) \neq 0$ a.e. have locally constant determinant except in a relatively closed set of measure zero, and that stationary points have constant determinant almost everywhere. This is deduced from a more general result concerning solutions $u \in Lip(Ω,\mathbb{R}^n)$, $Ω\subset \mathbb{R}^n$ to the linearized problem $curl(βDu) = 0$. We also present some generalization of the original result to higher dimensions and assuming further regularity on solutions $u$. Finally, we show that the differential inclusion associated to stationarity with respect to polyconvex energies as above is rigid.
